Given a tableau , let be a subset of the entries in the th column of , and let be a subset of the entries in the st column. We can come up with Garnir elements associated to this choice of and , but — as we pointed out last time — we need some way of picking which particular transversal elements to use. For each summand in , we separate into a pair of sets , but we have flexibility in how we order the elements of and . Our answer in this case is to always pick the permutation that puts the elements of into increasing order as we move down the columns of .
For example, consider the tableau
This tableau has a “row descent” in the second row: a pair of adjacent entries in the row where the larger entry is on the left instead of the right. Let be the entry on the left along with all the entries below it in its column — — and let be the entry on the right along with all the entries above it in its column — . We look at all six ways of rearranging the collection into two subsets of two elements each (we listed then last time, actually) and choose permutations that keep entries in increasing order as we move down the columns.
Notice that we’ve picked different permutations this time, and so we get a different Garnir element:
Also, note that only the first of these tableaux has the descent in the second row, although some now have descents in the first row. Slightly less obvious is the fact that , and so we can write
Thus we can rewrite this polytabloid that has a row descent in terms of a bunch of other polytabloids that don’t have it and are “more standard”, in a sense we’ll define later.
Let and be two disjoint sets of positive integers. We’re mostly interested in the symmetric group , which shuffles around all the integers in both sets. But a particularly interesting subgroup is , which shuffles around the integers in and , but doesn’t mix the two together. Clearly, is a subgroup of .
Now, let be a transversal collection of permutations for this subgroup. That is, we can decompose the group into cosets
Then a “Garnir element” is
Now, the problem here is that we’ve written as if it only depends on the sets and , when it clearly depends on the choice of the transversal . But we’ll leave this alone for the moment.
How can we come up with an explicit transversal in the first place? Well, consider the set of pairs of sets so that , , and . That is, each is another way of breaking the same collection of integers up into two parts of the same sizes as and .
Any permutation acts on the collection of such pairs of sets in the obvious way, sending to , which is another such pair. In fact, it’s transitive, since we can always find some with and . If for each we make just such a choice of , then this collection of permutations gives us a transversal!
We can check this by first making sure we have the right number of elements. A pair is determined by taking all integers and picking to go into . That is, we have
pairs. But this is also the number of cosets of in !
Do we accidentally get two representatives and for the same coset? If we did, then we’d have to have . But then , and thus . But we only picked one permutation sending to a given pair, so .
As an example, let and . Then we have six pairs of sets to consider
where in each case I’ve picked a that sends to . This will give us the following Garnir element:
But, again, this is far from the only possible choice for this and .
Now we’re all set to show that the polytabloids that come from standard tableaux are linearly independent. This is half of showing that they form a basis of our Specht modules. We’ll actually use a lemma that applies to any vector space with an ordered basis . Here indexes some set of basis vectors which has some partial order .
So, let be vectors in , and suppose that for each we can pick some basis vector which shows up with a nonzero coefficient in subject to the following two conditions. First, for each the basis element should be the maximum of all the basis vectors having nonzero coefficients in . Second, the are all distinct.
We should note that the first of these conditions actually places some restrictions on what vectors the can be in the first place. For each one, the collection of basis vectors with nonzero coefficients must have a maximum. That is, there must be some basis vector in the collection which is actually bigger (according to the partial order ) than all the others in the collection. It’s not sufficient for to be maximal, which only means that there is no larger index in the collection. The difference is similar to that between local maxima and a global maximum for a real-valued function.
This distinction should be kept in mind, since now we’re going to shuffle the order of the so that is maximal among the basis elements . That is, none of the other should be bigger than , although some may be incomparable with it. Now I say that cannot have a nonzero coefficient in any other of the . Indeed, if it had a nonzero coefficient in, say, , then by assumption we would have , which contradicts the maximality of . Thus in any linear combination
we must have , since there is no other way to cancel off all the occurrences of . Removing from the collection, we can repeat the reasoning with the remaining vectors until we get down to a single one, which is trivially independent.
So in the case we care about the space is the Young tabloid module , with the basis of Young tabloids having the dominance ordering. In particular, we consider for our the collection of polytabloids where is a standard tableau. In this case, we know that is the maximum of all the tabloids showing up as summands in . And these standard tabloids are all distinct, since they arise from distinct standard tableaux. Thus our lemma shows that not only are the standard polytabloids distinct, they are actually linearly independent vectors in .
Any such comes from , where . We will make our induction on the number of “column inversions” in . That is, the number of pairs of entries that are in the same column of , but which are “out of order”, in the sense that is in a lower row than .
Given any such pair, the dominance lemma tells us that . That is, by “untwisting” the column inversion, we can move up the dominance order while preserving the columns. It should also be clear that has fewer column inversions than does. But if we undo all the column inversions, the tableau we’re left with must be standard. That is, it must be itself.
If , and appears in a lower row than in the Young tabloid , then dominates . That is, swapping two entries of so as to move the lower number to a higher row moves the tabloid up in the dominance relations.
Let the composition sequences of and be and , respectively. For and we automatically have . For there is a difference between the two: the entry has been added in a different place. Let and be in rows and of , respectively. In , the entry is added to row , while in it’s been added to row . That is, is the same as with part increased by one and part decreased by one. Our assumption that is in a lower row than in is that . Therefore, since the lower row in is less than in , we find that . And we conclude that , as asserted.
Sorry, this should have gone up last Friday.
If is a Young tabloid with shape , we can define tabloids for each from to by letting be formed by the entries in less than or equal to . We define to be the shape of as a composition. For example, if we have
then we define
Along the way we see why we might want to consider a composition like with a zero part.
Anyway, now we define a dominance order on tabloids. If and are two tabloids with composition sequences and , respectively, then we say “dominates” — and we write — if dominates for all .
As a (big!) example, we can write down the dominance order on all tabloids of shape :
It’s an exercise to verify that these are indeed all the tabloids with this shape. For each arrow, we can verify the dominance. As an example, let’s show that
First, let’s write down their composition sequences:
Now it should be easy to see on each row that . As another example, let’s try to compare and . Again, we write down their composition sequences:
We see that , but . Thus neither tabloid dominates the other. The other examples to verify this diagram are all similarly straightforward.
A “composition” is sort of like a partition, except the parts are allowed to come in any specified order. That is, a composition of is an ordered sequence of nonnegative integers that sums up to . Every partition is a composition — specifically one in which the sequence is nonincreasing. Since a general composition allows its parts to increase, it’s possible that some of the are zero, which can’t really happen for partitions.
and one possible Young tableau with this shape is
Now, it should be made clear that this is not a standard tableau, despite the fact that the rows and columns increase. The usual line is that we imagine the tableau to be at the upper-left corner of a quarter-plane, so there are cells extending out to the right and bottom of the diagram that aren’t part of the tableau. These, we say, are all filled with , and so the second column of this particular tableau is “actually” , so it doesn’t actually increase after all. But as far as I can tell this is a lot of word salad designed to back up the definition we really want: the notion of standard Young tableaux only applies to partitions, not to compositions in general.
We can, however, extend the idea of the dominance order to general compositions. As usual we say that if
for all . We just don’t have to add up all the biggest parts first.
So we’ve described the Specht modules, and we’ve shown that they give us a complete set of irreducible representations for the symmetric groups. But we haven’t described them very explicitly, and we certainl can’t say much about them. There’s still work to be done.
Recall that we had a canonical Young tableau for each shape that listed the numbers from to in each row from top to bottom, as in
It should be clear that this canonical tableau is standard, so there is always at least one standard tableau for each shape. There may be more, of course. For example:
Clearly, any two distinct standard tableaux and give rise to distinct tabloids and . Indeed, if , then and would have to be row-equivalent. But only one Young tableau in any row-equivalence class has increasing rows, and only that one even has a chance to be standard. Thus if and are row-equivalent standard tableaux, they must be equal.
What’s not immediately clear is that the standard polytabloids and are distinct. Further, it turns out that the collection of standard polytabloids of shape is actually independent, and furnishes a basis for the Specht module . This is our next major goal.
We have a number of immediate consequences of the submodule theorem. First, and most important, the Specht modules form a complete list of irreducible modules for the symmetric group . We know that they’re irreducible, and that there’s one of them for each partition , which is the number of modules we’re looking for. But we need to show that the Specht modules corresponding to distinct partitions are themselves distinct. For this, we’ll use a lemma.
If is a nonzero intertwinor, then . Further, if , then must be multiplication by a scalar. Indeed, since there must be some polytabloid with . We decompose , and extent to all of by sending every vector in to . That is:
where the are -tableaux. Now, the can’t all be zero, so we must have at least one -tableau so that . But then our corollary of the sign lemma tells us that , as we asserted!
Further, if , then our other corollary shows us that for some scalar . We can thus calculate
and so multiplies every vector by .
As a consequence, the must be distinct for distinct permutations, since if then there is a nonzero homomorphism , and thus . But the same argument shows that , and thus .
More particularly, we have a decomposition
where the diagonal multiplicities are . The rest of these multiplicities will eventually have a nice interpretation.
Sorry for the delay.
Let be a submodule of one of the Young tabloid modules. Then I say that either contains the Specht module , or it is contained in the orthogonal complement . In particular, each Specht module is irreducible, since any nontrivial submodule cannot be contained in the orthogonal complement, and so it must also contain — and thus be equal to — .
To see this, let be any vector, and let be a -tableau. Our last result showed us that for some . First, we’ll assume that there is some pair of and so that . In this case, . Since generates all of — the Specht modules are cyclic — we must have .
On the other hand, what if there is no such pair of and ? That is, for all vectors and -tableaux . We can calculate
So each vector is orthogonal to each polytabloid . Since the polytabloids span , we must have .